A273514 a(n) is the number of arithmetic progressions m < n < p (three numbers in arithmetic progression) such that m and p contain no 2's in their ternary representation.

0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 8, 2, 2, 8, 8, 8, 8, 2, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2

Offset: 0

Author

Max Barrentine, May 23 2016

Keywords

• nonn
• base

Comments

This is a recursive sequence that gives the number of times n is rejected from A005836, if n is the middle member of an arithmetic triple whose first and last terms are contained in A005836.

Also, a(n) is the number of unordered pairs of members of A005836 whose average (arithmetic mean) is n.

It appears that when A273513(n) and A262097(n) are coprime, a(n) = 2.

Local maxima occur at a(A125857(n)).

Examples

			a(2) = 2 because there are two arithmetic triples a < 2 < b such that a and b are members of A005836: 0, 2, 4 and 1, 2, 3.
		

Links

• Max Barrentine, Table of n, a(n) for n = 0..19683

Crossrefs

Cf. A005836, A125857, A262097, A273513.

Programs

PARI

precCantor(n)=my(v=digits(n,3)); for(i=1,#v, if(v[i]==2, for(j=i,#v,v[j]=1); break)); fromdigits(v,2)
a(n)=if(n==0, return(0)); sum(i=0,precCantor(n-1), my(m=fromdigits(digits(i,2),3)); vecmax(digits(2*n-m,3))<2) \\ Charles R Greathouse IV, Jun 17 2016

Formula

a(0) = 0, a(n) = a(3n) = a(3n+1); if a(n) = 0, a(9n + 2) = 2, otherwise a(9n + 2) = 4a(n); a(9n + 5) = a(9n + 6) = a(9n + 7) = a(9n + 8) = a(3n + 2).